It depends on the material. For metal bands, I take the watch off and immerse the band and the spring bars in a 70% isopropyl alcohol solution for a few hours. This alcohol soak serves two purposes. First, the alcohol kills any bacteria present on the band, second, it loosens any impacted dirt and grime. Next, I remove the band from the alcohol, rinse it well with tap water, then use an old toothbrush with dish detergent to scrub the band. Rinse it again with tap water then dry it and put it back on the watch.
Fabric bands can be washed in a similar manner, skipping the alcohol soak. For leather, DO NOT do the alcohol soak as the alcohol will dry out the leather and it might cause other damage. For leather, I'll generally brush off any surface dirt, then rub in a good quality saddle soap.
I hope this helps.